An -ing ending adjective that doesn't have a base verb form

The answer might depend on whether you mean historically/diachronically, or in terms of whether the -ing word is currently/synchronically felt to be based on a related verb. If you are asking whether there is always a synchronic base, the adjective cunning seems to work as an answer to your first question. "(To) cun" does not exist as a verb in standard English. The adjective cunning is derived from an obsolete present participle form of the verb can (which once had an infinitive form built on the stem cunn-), but in modern English can inflects as a modal auxiliary and so lacks non-finite forms. Therefore, cunning cannot be interpreted as a present participle of can, and I think it would be incorrect to argue even for a derivational synchronic connection between can and cunning.

In response to your final question, linguists generally agree that not all -ing words in English are forms of a verb. An -ing word modified by very, such as exciting in "a very exciting day", is usually classified as an adjective that is lexically separate from the verb excite, rather than an inflected form of it. In contrast, exciting in a sentence like "The teachers should be exciting the students about learning" would be classified as an inflected form of the verb excite.

Some -ing adjectives are compounds without a corresponding single-word verb, like your example easy-going, which does not correspond to a single-word verb. Another example: the adjective everlasting exists, but *everlast is not in general use as a verb (Wiktionary records two examples that are negligible because of their rarity). Good-looking, which is used adjectivally, is spelled with a hyphen and could be categorized as a single, compound word, but *good-look is not in common use as a verb: instead, the two-word phrase look good is used (with look inflected into the appropriate form, e.g. "looking good"). Parts of these words are based on a verb, but the whole word is composed of more than just a verb plus the suffix -ing.